Method for voltage instability load shedding using local measurements

ABSTRACT

A method of voltage instability load shedding, that includes the steps of measuring current and voltage waveforms of an electrical system at a local bus estimating the Thevenin equivalent admittance based on Kalman Filter techniques, then a voltage stability margin index is calculated using the voltage magnitude. The determined Thevenin admittance and the load at the local system bus and the calculated voltage stability margin index is compared with a predetermined threshold value to determine whether to initiate a load shedding action.

TECHNICAL FIELD AND BACKGROUND OF THE INVENTION

This invention relates to a method of determining voltage stabilitymargin at local bus level and to applying the method to enhanceunder-voltage load shedding protection scheme. We named this newprotection scheme as “voltage instability load shedding”.

Under Voltage Load Shedding (UVLS) has been used as an economic means ofavoiding voltage collapse. Since load shedding results in high costs toelectricity suppliers and consumers, this option is only used when allother means of avoiding voltage collapse are exhausted. UVLS sheds loadin pre-defined blocks that are triggered in stages when local voltagedrops to the pre-defined levels.

In most UVLS schemes, voltage magnitude is the only triggeringcriterion. However, past research has demonstrated that voltagemagnitude alone is not a satisfactory indicator of the proximity tovoltage instability under all circumstances. In fact, voltage stabilityis determined by the ability of the power system to supply and deliverreactive power. In actual systems, the computation of actual system PVcurves may be very complicated due to the large number of generators,widespread applications of capacitor banks, uncertainty about thedynamic characteristics of system loads, and the variability of powerflow pattern. In addition, operation of under load tap changers, theactual dynamic reactive capability of generators and accurate reactivereserve all affect the ability of the system to supply and deliver thereactive power. Therefore, determination of proper settings for UVLSschemes becomes a challenging task for system planners.

Moreover, modeling uncertainties post more challenges for systemplanners to determine the proper settings for UVLS schemes. Currentsettings of UVLS are determined by system planning engineers throughextensive network analyses using computer simulation packages. However,simulated system behaviors do not usually coincide with actual measuredsystem responses due to data and modeling issues. Inappropriate settingscan result in unnecessary shedding or failure to detect the need forload shedding.

SUMMARY OF THE INVENTION

A new control method referred to as “Voltage Instability Load Shedding”(VILS) is disclosed in this application. This new control method canenhance the conventional UVLS at designated locations, such as majorload centers. This smart control scheme computes Voltage StabilityMargin Index (VSMI) continuously to track the voltage stability marginat local bus level. The VSMI is expressed as active, reactive, andapparent power. The VSMI is used as an adaptive triggering criterion forload shedding.

This VILS method comprises the steps of, or means for, measuring currentand voltage waveforms at the local load bus, therefrom estimatingThevenin equivalent admittance (Y), then calculating the VSMI, andfinally comparing the VSMI with the pre-set threshold to decide whetherto initiate a load shedding action.

Therefore, it is an object of the invention to provide a new loadshedding strategy subject to voltage instability.

It is another object of the invention to provide a new method ofestimating voltage stability margin at local bus level.

It is another object of the invention to express the voltage stabilitymargin in terms of real, reactive and apparent power.

BRIEF DESCRIPTION OF THE DRAWINGS

Some of the objects of the invention have been set forth above. Otherobjects and advantages of the invention will appear as the descriptionproceeds when taken in conjunction with the following drawings, inwhich:

FIG. 1 depicts the Thevenin equivalent system;

FIG. 2 depicts an exemplary graph where tracking closeness to voltageinstability becomes tracking the distance of the current load level tomaximum power transfer in terms of voltage stability limit; and

FIG. 3 depicts a flowchart of the operation of a VILS system.

DESCRIPTION OF THE PREFERRED EMBODIMENT AND BEST MODE

Referring now to FIG. 1, the real and reactive power transferred fromthe system to the load is

$\begin{matrix}\left\{ \begin{matrix}{P_{L} = {{{EVY}\; {\cos \left( {\alpha - \delta - \beta} \right)}} - {V^{2}G}}} \\{Q_{L} = {{{EVY}\; {\sin \left( {\alpha - \delta - \beta} \right)}} + {V^{2}B}}}\end{matrix} \right. & (1)\end{matrix}$

Where Y is the magnitude and β is the angle of the Thevenin equivalentadmittance G+jB

Dividing E²Y on both sides of Equation (1), it can be reformulated as:

$\begin{matrix}\left\{ {{{\begin{matrix}{p = {{v\; {\cos \left( {\alpha - \delta - \beta} \right)}} - {v^{2}\cos \; \beta}}} \\{q = {{v\; {\sin \left( {\alpha - \delta - \beta} \right)}} + {v^{2}\sin \; \beta}}}\end{matrix}{where}p} = \frac{P_{L}}{E^{2}Y}},{q = \frac{Q_{L}}{E^{2}Y}},{v = \frac{V}{E}}} \right. & (2)\end{matrix}$

Moving v² cos β and v² sin β to the left sides and taking the square ofthe right and left sides and adding, the following equation is obtained:

(p+v ² cos β)²+(q−v ² sin β)² =v ²  (3)

Substitute q with p·tan φ, where φ is the power factor of load.

From equation (3) is obtained:

p=−v ² cos φ cos(φ+β)+cos φ√{square root over (v ² −v ⁴ sin²(φ+β))}  (4)

Taking the derivative and setting it equal to zero, the normalizedcritical voltage and maximum power is obtained:

$\begin{matrix}{\frac{\partial p}{\partial v} = {{1 - {4v^{2}} + {4v^{4}{\sin^{2}\left( {\varphi + \beta} \right)}}} = 0}} & (5) \\{v_{critical}^{2} = {\frac{1 - {\cos \left( {\varphi + \beta} \right)}}{2\; {\sin^{2}\left( {\varphi + \beta} \right)}} = \frac{1}{2\left\lbrack {1 + {\cos \left( {\varphi + \beta} \right)}} \right\rbrack}}} & (6) \\{p_{\max} = \frac{\cos \; \varphi}{2\left\lbrack {1 + {\cos \left( {\varphi + \beta} \right)}} \right\rbrack}} & (7)\end{matrix}$

The maximum active and reactive transfer power is expressed as:

P _(max) =E ² Y·p _(max) =V ² Y cos φ

Q _(max) =E ² Y·Q _(max) =V ² Y sin φ  (8)

Therefore, as is shown in FIG. 2, tracking closeness to voltageinstability becomes tracking the distance of the current load level tomaximum power transfer. The VSMI expressed by equation (9-11) providesvoltage stability margin in terms of the apparent, active and reactivepower.

Voltage stability margin in terms of active power:

P _(Margin) =P _(max) −P _(L)  (9)

Voltage stability margin in terms of reactive power

Q _(Margin) =Q _(max) −Q _(L)  (10)

Voltage stability margin in terms of apparent power

S _(Margin)=√{square root over (P _(max) ² +Q _(max) ²)}−S _(L)  (11)

The closer the VSMI are to zero, the more imminent is the system tovoltage instability. VSMI also indicates how much load needs to be shedin order to prevent voltage instability.

In Equations (9-11), P_(L) and Q_(L) can be calculated using the localmeasurements of voltage and current samples, while the TheveninEquivalent admittance Y in (8) must be estimated. An estimation methodusing Kalman Filter is therefore developed.

The estimation equation is:

{circumflex over (z)}=H{circumflex over (x)}+{circumflex over (v)}  (12)

where {circumflex over (z)} is the measurement vector. {circumflex over(x)} is the state vector to be estimated, H is the observation model,and {circumflex over (v)} is the observation noise.

To minimize the estimation error, we are trying to minimize a costfunction that

$\begin{matrix}{J = {\frac{1}{2}\left( {\hat{z} - {H\hat{x}}} \right)^{T}\left( {\hat{z} - {H\hat{x}}} \right)}} & (13)\end{matrix}$

The criterion to minimize the J is that its derivative equals to zero

$\begin{matrix}{\frac{\partial J}{\partial\hat{x}} = {{{- \left( {\hat{z} - {H\hat{x}}} \right)^{T}}H} = 0}} & (14)\end{matrix}$

At that time the estimation of {circumflex over (x)} is given as

{circumflex over (x)} _(est)=(H ^(T) H)⁻¹ H ^(T) {circumflex over(z)}  (15)

Now deriving a recursive equation to estimate the {circumflex over (x)}.Let P is the covariance of the error in the estimator as:

$\begin{matrix}\begin{matrix}{P = {E\left\lbrack {{\overset{\sim}{x}}_{est}{\overset{\sim}{x}}_{est}^{T}} \right\rbrack}} \\{= {\left( {H^{T}H} \right)^{- 1}H^{T}{{RH}\left( {H^{T}H} \right)}^{- 1}}} \\{= \left( {H^{T}R^{- 1}H} \right)^{- 1}}\end{matrix} & (16)\end{matrix}$

where {tilde over (x)}=x−{tilde over (x)}_(est), and R is the covariancematrix of measurement error.

Represent equation (17) using the discrete measured values

$\begin{matrix}\begin{matrix}{P_{n} = \left( {\sum\limits_{i = 1}^{n}{H_{i}^{T}R_{i}^{- 1}H_{i}}} \right)^{- 1}} \\{= \left( {{\sum\limits_{i = 1}^{n - 1}{H_{i}^{T}R_{i}^{- 1}H_{i}}} + {H_{n}^{T}R_{n}^{- 1}H_{n}}} \right)^{- 1}} \\{= \left( {P_{n - 1}^{- 1} + {H_{n}^{T}R_{n}^{- 1}H_{n}}} \right)^{- 1}}\end{matrix} & (17)\end{matrix}$

Equation (15) at time instant n is written as:

$\begin{matrix}{\begin{matrix}{{\hat{x}}_{n} = {\left( {\sum\limits_{i = 1}^{n}{H_{i}^{T}R_{i}^{- 1}H_{i}}} \right)^{- 1}\left( {\sum\limits_{i = 1}^{n}{H_{i}^{T}R_{i}^{- 1}z_{i}}} \right)}} \\{= {P_{n}\left\lbrack {{\sum\limits_{i = 1}^{n - 1}{H_{i}^{T}R_{i}^{- 1}z_{i}}} + {H_{n}^{T}R_{n}^{- 1}z_{n}}} \right\rbrack}} \\{= {P_{n}\left( {{P_{n - 1}^{- 1}{\hat{x}}_{n - 1}} + {H_{n}^{T}R_{n}^{- 1}z_{n}}} \right)}}\end{matrix}{Define}} & (18) \\{K_{n} = {P_{n}H_{n}^{T}R_{n}^{- 1}}} & (19)\end{matrix}$

Then equation (18) would be

{circumflex over (x)} _(n) =P _(n) P _(n-1) ⁻¹ {circumflex over (x)}_(n-1) +K _(n) {circumflex over (z)} _(n)  (20)

Since

P _(n) P _(n-1) ⁻¹ =I−K _(n) H _(n)  (21)

Then, the recursive equation to estimate {circumflex over (x)}_(n) is

{circumflex over (x)} _(n) ={circumflex over (x)} _(n-1) +K _(n) [z _(n)−H _(n) {circumflex over (x)} _(n-1)]  (22)

Now apply the above method in the load shedding problem. From FIG. 1

E−jY ⁻¹ I=V  (23)

where

E is the Thevenin equivalent generator terminal voltage; V is the localload bus voltage; I is the line current; and Y is the Theveninequivalent admittance. V and I can be measured at local bus. DenoteE=E_(r)+jE_(i), V=m+jn, I=p+jq, Z=1/Y=R+jX.

Then, in accordance with equation (12):

$\begin{matrix}{\hat{z} = \begin{bmatrix}m \\n\end{bmatrix}} & (24) \\{H = \begin{bmatrix}1 & 0 & {- p} & q \\0 & 1 & {- q} & {- p}\end{bmatrix}} & (25) \\{\hat{x} = \begin{bmatrix}E_{r} \\E_{i} \\R \\X\end{bmatrix}} & (26)\end{matrix}$

When applying the recursive equation (22), several parameters need to beinitialized. According to the preferred embodiment:

-   -   (i) the initial value of {circumflex over (x)} is set based on        the power flow solution;    -   (ii) the covariance matrix of measurement error R, is set        according to the standard deviation of the measurement device,        which reflects the expected accuracy of the corresponding meter        used;    -   (iii) P is the covariance matrix of the estimator error. The        initial value of P is set as a diagonal matrix with the element        value equal to 0.000001.

The estimation method uses a sliding data window with four samples perwindow. The estimation of the Thevenin admittance is conductedcontinuously. Preferably, the sampling time step is set as 0.01 s or 1cycle based on 60 Hz. This sampling rate is determined based on theconsiderations of obtaining accurate estimation value of the Theveninadmittance and having enough time to detect a fault in order to blockthe load shedding function during the fault.

Referring now to the flowchart shown in FIG. 3, the proposed VILS methodis summarized. In FIG. 3, ε≧0 represents a mismatch margin that is setby the user.

The voltage and current samples are measured directly at local bus. Withthose samples, the active power of local load (P_(L)) and reactive powerof local load (Q_(L)) can be calculated and the Thevenin admittance Ycan be estimated using the Kalman Filter estimation method describedearlier. Then, voltage stability margin in terms of active power,reactive power and apparent power is calculated using (9˜11). Thismargin, then, compared with the user set mismatch margin, ε to determinewhether load shedding should be taken.

The voltage instability load shedding method is described above. Variousdetails of the invention may be changed without departing from itsscope. Furthermore, the foregoing description of the preferredembodiment of the invention and the best mode for practicing theinvention are provided for the purpose of illustration only and not forthe purpose of limitation—the invention being defined by the claims.

1. A method of voltage instability load shedding, comprising the steps of: (a) measuring current and voltage waveforms of an electrical system at a local system bus; (b) estimating the Thevenin equivalent admittance (Y) based on Kalman filter techniques; (c) calculating a voltage stability margin index using the magnitude of the voltage wave form, the determined Thevenin admittance, and the load at the local system bus; (d) comparing the calculated voltage stability margin index with a predetermined threshold value; and (e) utilizing the comparison between the calculated voltage stability margin index with the predetermined threshold value to determine whether to initiate a load shedding action.
 2. A method of voltage instability load shedding according to claim 1, wherein the Thevenin admittance is estimated based on the equation: {circumflex over (z)}=H{circumflex over (x)}+{circumflex over (v)} where {circumflex over (z)} is a measurement vector, {circumflex over (x)} is a state vector to be estimated, H is an observation model, {circumflex over (v)} is an observation noise.
 3. A method of voltage instability load shedding according to claim 2 and including the step wherein the initial value {circumflex over (x)} is set based on a power flow solution, a covariance matrix of measurement error R set according to the accuracy of the measurement device, and the initial value of the covariance of the estimation error P set as a diagonal matrix with the element value equal to 0.000001.
 4. A method of voltage instability load shedding according to claim 2, wherein the estimation of the Thevenin admittance is conducted continuously and the Kalman filter estimation method uses a sliding data window with four samples per window.
 5. A method of voltage instability load shedding according to claim 2, wherein the sampling time step is set as 1 cycle, based on 60 Hz.
 6. A method of voltage instability load shedding according to claim 2, wherein the sampling rate is determined based on the estimation value of the Thevenin admittance with sufficient time to detect a fault in order to block the load shedding function during the fault.
 7. A method of voltage instability load shedding according to claim 1, wherein the voltage stability margin is expressed in terms of an apparent, active and reactive power. 